Nonlinear Fiber Optics - 4 ed. Agrawal

3.3. Third-Order Dispersion 65
Figure 3.7: Evolution of a super-Gaussian pulse with m = 3 along the fiber length for the case
of β 2 = 0 and β 3 > 0. Third-order dispersion is responsible for the oscillatory structure near the
trailing edge of the pulse.
the FWHM is not a true measure of the width of pulses shown in Figures 3.6 and 3.7,
we use the RMS width σ defined in Eq. (3.2.26). In the case of Gaussian pulses, it is
possible to obtain a simple analytic expression of σ that includes the effects of β 2 , β 3 ,
and the initial chirp C on dispersion broadening [9].
3.3.2 Broadening Factor
To calculate σ from Eq. (3.2.26), we need to find the nth moment 〈T n 〉 of T using Eq.
(3.2.27). As the Fourier transform Ũ(z,ω) ofU(z,T ) is known from Eq. (3.3.2), it is
useful to evaluate 〈T n 〉 in the frequency domain. By using the Fourier transform Ĩ(z,ω)
of the pulse intensity |U(z,T )| 2 ,
∫ ∞
Ĩ(z,ω)= |U(z,T )| 2 exp(iωT )dT, (3.3.7)
−∞
and differentiating it n times, we obtain
lim
ω→0
∂ n
∂ω n Ĩ(z,ω)=(i)n ∫ ∞
Using Eq. (3.3.8) in Eq. (3.2.27) we find that
where the normalization constant
N c =
〈T n 〉 = (−i)n
N c
∫ ∞
−∞
|U(z,T )| 2 dT ≡
−∞
∂ n
T n |U(z,T )| 2 dT. (3.3.8)
lim Ĩ(z,ω), (3.3.9)
ω→0 ∂ωn ∫ ∞
−∞
|U(0,T )| 2 dT. (3.3.10)